Two tiny conducting spheres are identical and carry charges of -19.8 µC and +54.5 µC. They are separated by a distance of 2.45 cm.
(a) What is the magnitude of the force that each sphere experiences, and is the force attractive or repulsive?
1 N 2 ---Select--- attractive repulsive
(b) The spheres are brought into contact and then separated to a distance of 2.45 cm. Determine the magnitude of the force that each sphere now experiences, and state whether the force is attractive or repulsive.
To find the magnitude of the force between the two conducting spheres, we can use Coulomb's Law. Coulomb's Law states that the force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them.
(a) Given:
- Charge on one sphere = -19.8 µC
- Charge on the other sphere = +54.5 µC
- Distance between the spheres = 2.45 cm
First, we need to convert the charges from microcoulombs (µC) to coulombs (C):
- Charge on one sphere = -19.8 × 10^-6 C
- Charge on the other sphere = +54.5 × 10^-6 C
Now, we can calculate the magnitude of the force using Coulomb's Law:
F = (k * |q1 * q2|) / r^2
where:
- F is the magnitude of the force
- k is the electrostatic constant (k = 9 × 10^9 N·m^2/C^2)
- q1 and q2 are the charges on the spheres
- r is the distance between the spheres
Let's plug in the values:
F = (9 × 10^9 N·m^2/C^2 * |(-19.8 × 10^-6 C) * (+54.5 × 10^-6 C)|) / (0.0245 m)^2
Now we can solve this expression:
F = (9 × 10^9 N·m^2/C^2 * |(-19.8 * 54.5) × 10^-6 C^2|) / (0.0245^2 m^2)
F = (9 × 10^9 N·m^2/C^2 * 1078.1 × 10^-6 C^2) / 0.0006 m^2
F = (9 × 10^9 N·m^2 * 1078.1 × 10^-6 C^2) / 0.0006 m^2
F = 12.168 N
The magnitude of the force between the two spheres is 12.168 N. Now, we need to determine whether the force is attractive or repulsive. Since one sphere has a positive charge and the other has a negative charge, the force between them is attractive.
Therefore, the answer to part (a) is:
- Magnitude of the force = 12.168 N
- Force is attractive.
(b) After the spheres are brought into contact and then separated again to a distance of 2.45 cm, the charges distribute themselves evenly across both spheres. This means that the charge is now evenly split between the two spheres.
Since the charges are equal (+54.5 µC / 2 = +27.25 µC and -19.8 µC / 2 = -9.9 µC), the magnitude of the force can be calculated using the same steps as in part (a), but with the new charges:
F = (9 × 10^9 N·m^2/C^2 * |(-9.9 × 10^-6 C) * (+27.25 × 10^-6 C)|) / (0.0245 m)^2
F = (9 × 10^9 N·m^2 * 269.48 × 10^-6 C^2) / 0.0006 m^2
F = 0.3952 N
The magnitude of the force between the two spheres after contact and separation is 0.3952 N.
Since the charges are now the same sign (both positive), the force between the spheres is repulsive.
Therefore, the answer to part (b) is:
- Magnitude of the force = 0.3952 N
- Force is repulsive.